Question

A large rocky planet was formed from eight small rocky planets, each with radius $R$. Each of the small planets and the new large planet have the same density. The small rocky planets were initially so far apart that they didn't interact.
How many times as much energy would be required to blast the large planet apart as to blast all of the eight small planets individually apart?
(Consider that the energy required to blast a small planet of mass $m$ and radius $r$ apart is $\frac{3G{m}^2}{5r}$, where $G$ is a constant.)

• A. Half as much
• B. The same amount
• C. Twice as much
• D. $4$ times as much

Answer 1

The correct option is D $4$ times as much

As the density is constant $m\propto r^3$.
Mass of the large planet is $M=8m$
Radius of the larger planet is $R=\sqrt[3]{38}r=2r$

The total energy required to blast apart the large planet is then $\left(\frac{3G{M}^2}{5R}\right)=\frac{3G(8m{)}^2}{10r}=\left(32\right)\frac{3G{m}^2}{5r}$.
The sum of the energies for the 8 small planets is $8\left(\frac{3G{m}^2}{5r}\right)$,
which is $4$ times less than the large planet.